LMFDB - Embedded newform 800.4.c.m.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2068430400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 23x^{4} + 133x^{2} + 36 $ x^6 + 23x^4 + 133x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-3.12873i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.m.449.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.25747i q^{3} +9.15590i q^{7} -0.640965 q^{9} +O(q^{10})$	$q-5.25747i q^{3} +9.15590i q^{7} -0.640965 q^{9} -11.8984 q^{11} +41.9826i q^{13} -75.7604i q^{17} -2.49047 q^{19} +48.1368 q^{21} +17.8566i q^{23} -138.582i q^{27} +143.538 q^{29} -88.6094 q^{31} +62.5556i q^{33} -351.059i q^{37} +220.722 q^{39} +195.922 q^{41} -366.067i q^{43} +58.5931i q^{47} +259.169 q^{49} -398.308 q^{51} -0.374917i q^{53} +13.0936i q^{57} -318.952 q^{59} +446.597 q^{61} -5.86861i q^{63} -709.343i q^{67} +93.8805 q^{69} -1137.43 q^{71} -85.3855i q^{73} -108.941i q^{77} +1249.83 q^{79} -745.895 q^{81} -926.261i q^{83} -754.648i q^{87} -973.552 q^{89} -384.389 q^{91} +465.861i q^{93} +1158.55i q^{97} +7.62648 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 36 q^{9}+O(q^{10}) $ 6 * q - 36 * q^9	$ 6 q - 36 q^{9} - 62 q^{11} + 174 q^{19} - 140 q^{21} - 168 q^{29} - 588 q^{31} - 64 q^{39} - 690 q^{41} - 718 q^{49} - 2702 q^{51} + 2080 q^{59} + 964 q^{61} + 1228 q^{69} - 4096 q^{71} + 1996 q^{79} - 1098 q^{81} - 2378 q^{89} - 8064 q^{91} + 3908 q^{99}+O(q^{100}) $ 6 * q - 36 * q^9 - 62 * q^11 + 174 * q^19 - 140 * q^21 - 168 * q^29 - 588 * q^31 - 64 * q^39 - 690 * q^41 - 718 * q^49 - 2702 * q^51 + 2080 * q^59 + 964 * q^61 + 1228 * q^69 - 4096 * q^71 + 1996 * q^79 - 1098 * q^81 - 2378 * q^89 - 8064 * q^91 + 3908 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 5.25747i	− 1.01180i	−0.862592	−	0.505900i	$-0.831160\pi$
$3$	− 5.25747i	− 1.01180i	0.862592	−	0.505900i	$-0.168840\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.15590i	0.494372i	0.968968	+	0.247186i	$0.0795059\pi$
$7$	9.15590i	0.494372i	−0.968968	+	0.247186i	$0.920494\pi$
$8$	0	0
$9$	−0.640965	−0.0237394
$10$	0	0
$11$	−11.8984	−0.326137	−0.163069	−	0.986615i	$-0.552139\pi$
$11$	−11.8984	−0.326137	−0.163069	+	0.986615i	$0.552139\pi$
$12$	0	0
$13$	41.9826i	0.895684i	0.894113	+	0.447842i	$0.147807\pi$
$13$	41.9826i	0.895684i	−0.894113	+	0.447842i	$0.852193\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 75.7604i	− 1.08086i	−0.841389	−	0.540430i	$-0.818262\pi$
$17$	− 75.7604i	− 1.08086i	0.841389	−	0.540430i	$-0.181738\pi$
$18$	0	0
$19$	−2.49047	−0.0300712	−0.0150356	−	0.999887i	$-0.504786\pi$
$19$	−2.49047	−0.0300712	−0.0150356	+	0.999887i	$0.504786\pi$
$20$	0	0
$21$	48.1368	0.500206
$22$	0	0
$23$	17.8566i	0.161885i	0.996719	+	0.0809426i	$0.0257930\pi$
$23$	17.8566i	0.161885i	−0.996719	+	0.0809426i	$0.974207\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 138.582i	− 0.987781i
$28$	0	0
$29$	143.538	0.919117	0.459558	−	0.888148i	$-0.348008\pi$
$29$	143.538	0.919117	0.459558	+	0.888148i	$0.348008\pi$
$30$	0	0
$31$	−88.6094	−0.513378	−0.256689	−	0.966494i	$-0.582632\pi$
$31$	−88.6094	−0.513378	−0.256689	+	0.966494i	$0.582632\pi$
$32$	0	0
$33$	62.5556i	0.329986i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 351.059i	− 1.55983i	−0.625884	−	0.779916i	$-0.715262\pi$
$37$	− 351.059i	− 1.55983i	0.625884	−	0.779916i	$-0.284738\pi$
$38$	0	0
$39$	220.722	0.906253
$40$	0	0
$41$	195.922	0.746291	0.373145	−	0.927773i	$-0.378279\pi$
$41$	195.922	0.746291	0.373145	+	0.927773i	$0.378279\pi$
$42$	0	0
$43$	− 366.067i	− 1.29825i	−0.760682	−	0.649125i	$-0.775135\pi$
$43$	− 366.067i	− 1.29825i	0.760682	−	0.649125i	$-0.224865\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	58.5931i	0.181844i	0.995858	+	0.0909222i	$0.0289815\pi$
$47$	58.5931i	0.181844i	−0.995858	+	0.0909222i	$0.971019\pi$
$48$	0	0
$49$	259.169	0.755596
$50$	0	0
$51$	−398.308	−1.09361
$52$	0	0
$53$	− 0.374917i	0 0.000971675i	−1.00000		0.000485837i	$-0.999845\pi$
$53$	− 0.374917i	0 0.000971675i	1.00000		0.000485837i	$-0.000154647\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.0936i	0.0304261i
$58$	0	0
$59$	−318.952	−0.703796	−0.351898	−	0.936038i	$-0.614464\pi$
$59$	−318.952	−0.703796	−0.351898	+	0.936038i	$0.614464\pi$
$60$	0	0
$61$	446.597	0.937391	0.468695	−	0.883360i	$-0.344724\pi$
$61$	446.597	0.937391	0.468695	+	0.883360i	$0.344724\pi$
$62$	0	0
$63$	− 5.86861i	− 0.0117361i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 709.343i	− 1.29343i	−0.762730	−	0.646717i	$-0.776142\pi$
$67$	− 709.343i	− 1.29343i	0.762730	−	0.646717i	$-0.223858\pi$
$68$	0	0
$69$	93.8805	0.163795
$70$	0	0
$71$	−1137.43	−1.90124	−0.950621	−	0.310354i	$-0.899552\pi$
$71$	−1137.43	−1.90124	−0.950621	+	0.310354i	$0.899552\pi$
$72$	0	0
$73$	− 85.3855i	− 0.136899i	−0.997655	−	0.0684495i	$-0.978195\pi$
$73$	− 85.3855i	− 0.136899i	0.997655	−	0.0684495i	$-0.0218052\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 108.941i	− 0.161233i
$78$	0	0
$79$	1249.83	1.77996	0.889981	−	0.455997i	$-0.150717\pi$
$79$	1249.83	1.77996	0.889981	+	0.455997i	$0.150717\pi$
$80$	0	0
$81$	−745.895	−1.02318
$82$	0	0
$83$	− 926.261i	− 1.22494i	−0.790492	−	0.612472i	$-0.790175\pi$
$83$	− 926.261i	− 1.22494i	0.790492	−	0.612472i	$-0.209825\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 754.648i	− 0.929962i
$88$	0	0
$89$	−973.552	−1.15951	−0.579755	−	0.814791i	$-0.696852\pi$
$89$	−973.552	−1.15951	−0.579755	+	0.814791i	$0.696852\pi$
$90$	0	0
$91$	−384.389	−0.442801
$92$	0	0
$93$	465.861i	0.519436i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1158.55i	1.21271i	0.795194	+	0.606355i	$0.207369\pi$
$97$	1158.55i	1.21271i	−0.795194	+	0.606355i	$0.792631\pi$
$98$	0	0
$99$	7.62648	0.00774232
$100$	0	0
$101$	−466.816	−0.459900	−0.229950	−	0.973202i	$-0.573856\pi$
$101$	−466.816	−0.459900	−0.229950	+	0.973202i	$0.573856\pi$
$102$	0	0
$103$	− 963.139i	− 0.921368i	−0.887564	−	0.460684i	$-0.847604\pi$
$103$	− 963.139i	− 0.921368i	0.887564	−	0.460684i	$-0.152396\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1066.94i	− 0.963971i	−0.876179	−	0.481986i	$-0.839916\pi$
$107$	− 1066.94i	− 0.963971i	0.876179	−	0.481986i	$-0.160084\pi$
$108$	0	0
$109$	217.521	0.191145	0.0955723	−	0.995422i	$-0.469532\pi$
$109$	217.521	0.191145	0.0955723	+	0.995422i	$0.469532\pi$
$110$	0	0
$111$	−1845.68	−1.57824
$112$	0	0
$113$	− 1160.71i	− 0.966286i	−0.875542	−	0.483143i	$-0.839495\pi$
$113$	− 1160.71i	− 0.966286i	0.875542	−	0.483143i	$-0.160505\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 26.9094i	− 0.0212630i
$118$	0	0
$119$	693.655	0.534347
$120$	0	0
$121$	−1189.43	−0.893634
$122$	0	0
$123$	− 1030.06i	− 0.755097i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	34.2804i	0.0239519i	0.999928	+	0.0119759i	$0.00381215\pi$
$127$	34.2804i	0.0239519i	−0.999928	+	0.0119759i	$0.996188\pi$
$128$	0	0
$129$	−1924.59	−1.31357
$130$	0	0
$131$	−2364.02	−1.57669	−0.788343	−	0.615236i	$-0.789061\pi$
$131$	−2364.02	−1.57669	−0.788343	+	0.615236i	$0.789061\pi$
$132$	0	0
$133$	− 22.8025i	− 0.0148664i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2991.88i	1.86579i	0.360144	+	0.932897i	$0.382728\pi$
$137$	2991.88i	1.86579i	−0.360144	+	0.932897i	$0.617272\pi$
$138$	0	0
$139$	132.246	0.0806974	0.0403487	−	0.999186i	$-0.487153\pi$
$139$	132.246	0.0806974	0.0403487	+	0.999186i	$0.487153\pi$
$140$	0	0
$141$	308.052	0.183990
$142$	0	0
$143$	− 499.528i	− 0.292116i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1362.58i	− 0.764512i
$148$	0	0
$149$	788.190	0.433363	0.216681	−	0.976242i	$-0.430477\pi$
$149$	788.190	0.433363	0.216681	+	0.976242i	$0.430477\pi$
$150$	0	0
$151$	−2200.74	−1.18605	−0.593025	−	0.805184i	$-0.702066\pi$
$151$	−2200.74	−1.18605	−0.593025	+	0.805184i	$0.702066\pi$
$152$	0	0
$153$	48.5598i	0.0256590i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1702.05i	− 0.865212i	−0.901583	−	0.432606i	$-0.857594\pi$
$157$	− 1702.05i	− 0.865212i	0.901583	−	0.432606i	$-0.142406\pi$
$158$	0	0
$159$	−1.97111	−0.000983140 0
$160$	0	0
$161$	−163.493	−0.0800315
$162$	0	0
$163$	− 822.009i	− 0.394998i	−0.980303	−	0.197499i	$-0.936718\pi$
$163$	− 822.009i	− 0.394998i	0.980303	−	0.197499i	$-0.0632819\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2545.32i	− 1.17942i	−0.807615	−	0.589710i	$-0.799242\pi$
$167$	− 2545.32i	− 1.17942i	0.807615	−	0.589710i	$-0.200758\pi$
$168$	0	0
$169$	434.458	0.197751
$170$	0	0
$171$	1.59631	0.000713874 0
$172$	0	0
$173$	− 868.181i	− 0.381541i	−0.981635	−	0.190770i	$-0.938901\pi$
$173$	− 868.181i	− 0.381541i	0.981635	−	0.190770i	$-0.0610986\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1676.88i	0.712101i
$178$	0	0
$179$	3255.65	1.35944	0.679718	−	0.733474i	$-0.262102\pi$
$179$	3255.65	1.35944	0.679718	+	0.733474i	$0.262102\pi$
$180$	0	0
$181$	3482.49	1.43012	0.715059	−	0.699065i	$-0.246400\pi$
$181$	3482.49	1.43012	0.715059	+	0.699065i	$0.246400\pi$
$182$	0	0
$183$	− 2347.97i	− 0.948452i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	901.431i	0.352509i
$188$	0	0
$189$	1268.84	0.488331
$190$	0	0
$191$	2981.37	1.12945	0.564724	−	0.825280i	$-0.308983\pi$
$191$	2981.37	1.12945	0.564724	+	0.825280i	$0.308983\pi$
$192$	0	0
$193$	− 1666.53i	− 0.621552i	−0.950483	−	0.310776i	$-0.899411\pi$
$193$	− 1666.53i	− 0.621552i	0.950483	−	0.310776i	$-0.100589\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4355.01i	− 1.57503i	−0.616294	−	0.787516i	$-0.711367\pi$
$197$	− 4355.01i	− 1.57503i	0.616294	−	0.787516i	$-0.288633\pi$
$198$	0	0
$199$	−820.282	−0.292202	−0.146101	−	0.989270i	$-0.546672\pi$
$199$	−820.282	−0.292202	−0.146101	+	0.989270i	$0.546672\pi$
$200$	0	0
$201$	−3729.35	−1.30870
$202$	0	0
$203$	1314.22i	0.454386i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 11.4455i	− 0.00384306i
$208$	0	0
$209$	29.6327	0.00980736
$210$	0	0
$211$	−4678.00	−1.52629	−0.763144	−	0.646229i	$-0.776345\pi$
$211$	−4678.00	−1.52629	−0.763144	+	0.646229i	$0.776345\pi$
$212$	0	0
$213$	5980.00i	1.92368i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 811.299i	− 0.253800i
$218$	0	0
$219$	−448.912	−0.138514
$220$	0	0
$221$	3180.62	0.968108
$222$	0	0
$223$	2238.14i	0.672094i	0.941845	+	0.336047i	$0.109090\pi$
$223$	2238.14i	0.672094i	−0.941845	+	0.336047i	$0.890910\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1036.89i	0.303175i	0.988444	+	0.151588i	$0.0484386\pi$
$227$	1036.89i	0.303175i	−0.988444	+	0.151588i	$0.951561\pi$
$228$	0	0
$229$	3888.44	1.12208	0.561038	−	0.827790i	$-0.310402\pi$
$229$	3888.44	1.12208	0.561038	+	0.827790i	$0.310402\pi$
$230$	0	0
$231$	−572.753	−0.163136
$232$	0	0
$233$	− 130.042i	− 0.0365638i	−0.999833	−	0.0182819i	$-0.994180\pi$
$233$	− 130.042i	− 0.0365638i	0.999833	−	0.0182819i	$-0.00581963\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6570.95i	− 1.80097i
$238$	0	0
$239$	−5724.03	−1.54919	−0.774595	−	0.632458i	$-0.782046\pi$
$239$	−5724.03	−1.54919	−0.774595	+	0.632458i	$0.782046\pi$
$240$	0	0
$241$	1818.75	0.486123	0.243062	−	0.970011i	$-0.421848\pi$
$241$	1818.75	0.486123	0.243062	+	0.970011i	$0.421848\pi$
$242$	0	0
$243$	179.812i	0.0474689i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 104.557i	− 0.0269343i
$248$	0	0
$249$	−4869.79	−1.23940
$250$	0	0
$251$	3637.78	0.914800	0.457400	−	0.889261i	$-0.348781\pi$
$251$	3637.78	0.914800	0.457400	+	0.889261i	$0.348781\pi$
$252$	0	0
$253$	− 212.466i	− 0.0527968i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2969.73i	0.720804i	0.932797	+	0.360402i	$0.117360\pi$
$257$	2969.73i	0.720804i	−0.932797	+	0.360402i	$0.882640\pi$
$258$	0	0
$259$	3214.26	0.771137
$260$	0	0
$261$	−92.0030	−0.0218193
$262$	0	0
$263$	− 3035.72i	− 0.711752i	−0.934533	−	0.355876i	$-0.884183\pi$
$263$	− 3035.72i	− 0.711752i	0.934533	−	0.355876i	$-0.115817\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5118.42i	1.17319i
$268$	0	0
$269$	6888.55	1.56135	0.780673	−	0.624939i	$-0.214876\pi$
$269$	6888.55	1.56135	0.780673	+	0.624939i	$0.214876\pi$
$270$	0	0
$271$	6173.52	1.38382	0.691909	−	0.721985i	$-0.256770\pi$
$271$	6173.52	1.38382	0.691909	+	0.721985i	$0.256770\pi$
$272$	0	0
$273$	2020.91i	0.448026i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 3273.30i	− 0.710013i	−0.934864	−	0.355007i	$-0.884479\pi$
$277$	− 3273.30i	− 0.710013i	0.934864	−	0.355007i	$-0.115521\pi$
$278$	0	0
$279$	56.7955	0.0121873
$280$	0	0
$281$	458.400	0.0973161	0.0486581	−	0.998815i	$-0.484506\pi$
$281$	458.400	0.0973161	0.0486581	+	0.998815i	$0.484506\pi$
$282$	0	0
$283$	8064.66i	1.69397i	0.531614	+	0.846987i	$0.321586\pi$
$283$	8064.66i	1.69397i	−0.531614	+	0.846987i	$0.678414\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1793.84i	0.368945i
$288$	0	0
$289$	−826.645	−0.168257
$290$	0	0
$291$	6091.03	1.22702
$292$	0	0
$293$	812.863i	0.162075i	0.996711	+	0.0810375i	$0.0258234\pi$
$293$	812.863i	0.162075i	−0.996711	+	0.0810375i	$0.974177\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1648.91i	0.322152i
$298$	0	0
$299$	−749.667	−0.144998
$300$	0	0
$301$	3351.68	0.641819
$302$	0	0
$303$	2454.27i	0.465327i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	383.469i	0.0712891i	0.999365	+	0.0356445i	$0.0113484\pi$
$307$	383.469i	0.0712891i	−0.999365	+	0.0356445i	$0.988652\pi$
$308$	0	0
$309$	−5063.67	−0.932240
$310$	0	0
$311$	1665.18	0.303613	0.151807	−	0.988410i	$-0.451491\pi$
$311$	1665.18	0.303613	0.151807	+	0.988410i	$0.451491\pi$
$312$	0	0
$313$	− 3769.02i	− 0.680632i	−0.940311	−	0.340316i	$-0.889466\pi$
$313$	− 3769.02i	− 0.680632i	0.940311	−	0.340316i	$-0.110534\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1859.70i	0.329498i	0.986336	+	0.164749i	$0.0526814\pi$
$317$	1859.70i	0.329498i	−0.986336	+	0.164749i	$0.947319\pi$
$318$	0	0
$319$	−1707.88	−0.299758
$320$	0	0
$321$	−5609.40	−0.975346
$322$	0	0
$323$	188.679i	0.0325028i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1143.61i	− 0.193400i
$328$	0	0
$329$	−536.473	−0.0898988
$330$	0	0
$331$	7235.06	1.20144	0.600718	−	0.799461i	$-0.294882\pi$
$331$	7235.06	1.20144	0.600718	+	0.799461i	$0.294882\pi$
$332$	0	0
$333$	225.017i	0.0370295i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9890.08i	1.59866i	0.600895	+	0.799328i	$0.294811\pi$
$337$	9890.08i	1.59866i	−0.600895	+	0.799328i	$0.705189\pi$
$338$	0	0
$339$	−6102.39	−0.977688
$340$	0	0
$341$	1054.31	0.167432
$342$	0	0
$343$	5513.40i	0.867918i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6269.88i	0.969984i	0.874519	+	0.484992i	$0.161178\pi$
$347$	6269.88i	0.969984i	−0.874519	+	0.484992i	$0.838822\pi$
$348$	0	0
$349$	2436.94	0.373772	0.186886	−	0.982382i	$-0.440160\pi$
$349$	2436.94	0.373772	0.186886	+	0.982382i	$0.440160\pi$
$350$	0	0
$351$	5818.03	0.884739
$352$	0	0
$353$	− 10712.4i	− 1.61519i	−0.589738	−	0.807595i	$-0.700769\pi$
$353$	− 10712.4i	− 1.61519i	0.589738	−	0.807595i	$-0.299231\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3646.87i	− 0.540652i
$358$	0	0
$359$	−3540.39	−0.520487	−0.260243	−	0.965543i	$-0.583803\pi$
$359$	−3540.39	−0.520487	−0.260243	+	0.965543i	$0.583803\pi$
$360$	0	0
$361$	−6852.80	−0.999096
$362$	0	0
$363$	6253.38i	0.904179i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12208.3i	1.73642i	0.496193	+	0.868212i	$0.334731\pi$
$367$	12208.3i	1.73642i	−0.496193	+	0.868212i	$0.665269\pi$
$368$	0	0
$369$	−125.579	−0.0177165
$370$	0	0
$371$	3.43270	0.000480369 0
$372$	0	0
$373$	− 3056.65i	− 0.424309i	−0.977236	−	0.212154i	$-0.931952\pi$
$373$	− 3056.65i	− 0.424309i	0.977236	−	0.212154i	$-0.0680480\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6026.11i	0.823238i
$378$	0	0
$379$	−1476.91	−0.200168	−0.100084	−	0.994979i	$-0.531911\pi$
$379$	−1476.91	−0.200168	−0.100084	+	0.994979i	$0.531911\pi$
$380$	0	0
$381$	180.228	0.0242345
$382$	0	0
$383$	5567.95i	0.742843i	0.928464	+	0.371422i	$0.121130\pi$
$383$	5567.95i	0.742843i	−0.928464	+	0.371422i	$0.878870\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	234.636i	0.0308197i
$388$	0	0
$389$	−5401.03	−0.703966	−0.351983	−	0.936006i	$-0.614493\pi$
$389$	−5401.03	−0.703966	−0.351983	+	0.936006i	$0.614493\pi$
$390$	0	0
$391$	1352.82	0.174975
$392$	0	0
$393$	12428.8i	1.59529i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8592.52i	1.08626i	0.839648	+	0.543131i	$0.182761\pi$
$397$	8592.52i	1.08626i	−0.839648	+	0.543131i	$0.817239\pi$
$398$	0	0
$399$	−119.884	−0.0150418
$400$	0	0
$401$	−1794.62	−0.223489	−0.111745	−	0.993737i	$-0.535644\pi$
$401$	−1794.62	−0.223489	−0.111745	+	0.993737i	$0.535644\pi$
$402$	0	0
$403$	− 3720.06i	− 0.459824i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4177.05i	0.508720i
$408$	0	0
$409$	−11342.3	−1.37125	−0.685623	−	0.727957i	$-0.740470\pi$
$409$	−11342.3	−1.37125	−0.685623	+	0.727957i	$0.740470\pi$
$410$	0	0
$411$	15729.7	1.88781
$412$	0	0
$413$	− 2920.29i	− 0.347937i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 695.278i	− 0.0816496i
$418$	0	0
$419$	−8568.54	−0.999047	−0.499524	−	0.866300i	$-0.666492\pi$
$419$	−8568.54	−0.999047	−0.499524	+	0.866300i	$0.666492\pi$
$420$	0	0
$421$	−7705.53	−0.892030	−0.446015	−	0.895026i	$-0.647157\pi$
$421$	−7705.53	−0.892030	−0.446015	+	0.895026i	$0.647157\pi$
$422$	0	0
$423$	− 37.5561i	− 0.00431688i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4088.99i	0.463420i
$428$	0	0
$429$	−2626.25	−0.295563
$430$	0	0
$431$	12316.2	1.37645	0.688226	−	0.725496i	$-0.258390\pi$
$431$	12316.2	1.37645	0.688226	+	0.725496i	$0.258390\pi$
$432$	0	0
$433$	− 7105.69i	− 0.788632i	−0.918975	−	0.394316i	$-0.870981\pi$
$433$	− 7105.69i	− 0.788632i	0.918975	−	0.394316i	$-0.129019\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 44.4714i	− 0.00486809i
$438$	0	0
$439$	17585.8	1.91190	0.955951	−	0.293527i	$-0.0948292\pi$
$439$	17585.8	1.91190	0.955951	+	0.293527i	$0.0948292\pi$
$440$	0	0
$441$	−166.119	−0.0179374
$442$	0	0
$443$	− 2154.24i	− 0.231041i	−0.993305	−	0.115520i	$-0.963146\pi$
$443$	− 2154.24i	− 0.231041i	0.993305	−	0.115520i	$-0.0368536\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4143.88i	− 0.438476i
$448$	0	0
$449$	6853.94	0.720395	0.360197	−	0.932876i	$-0.382709\pi$
$449$	6853.94	0.720395	0.360197	+	0.932876i	$0.382709\pi$
$450$	0	0
$451$	−2331.17	−0.243393
$452$	0	0
$453$	11570.3i	1.20005i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1822.27i	− 0.186526i	−0.995642	−	0.0932629i	$-0.970270\pi$
$457$	− 1822.27i	− 0.186526i	0.995642	−	0.0932629i	$-0.0297297\pi$
$458$	0	0
$459$	−10499.0	−1.06765
$460$	0	0
$461$	3510.72	0.354687	0.177343	−	0.984149i	$-0.443250\pi$
$461$	3510.72	0.354687	0.177343	+	0.984149i	$0.443250\pi$
$462$	0	0
$463$	13454.1i	1.35047i	0.737603	+	0.675235i	$0.235958\pi$
$463$	13454.1i	1.35047i	−0.737603	+	0.675235i	$0.764042\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12549.1i	− 1.24347i	−0.783227	−	0.621736i	$-0.786428\pi$
$467$	− 12549.1i	− 1.24347i	0.783227	−	0.621736i	$-0.213572\pi$
$468$	0	0
$469$	6494.67	0.639438
$470$	0	0
$471$	−8948.46	−0.875421
$472$	0	0
$473$	4355.63i	0.423408i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.240308i	0 2.30670e-5i
$478$	0	0
$479$	−6582.21	−0.627868	−0.313934	−	0.949445i	$-0.601647\pi$
$479$	−6582.21	−0.627868	−0.313934	+	0.949445i	$0.601647\pi$
$480$	0	0
$481$	14738.4	1.39712
$482$	0	0
$483$	859.561i	0.0809759i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10288.5i	0.957326i	0.877999	+	0.478663i	$0.158878\pi$
$487$	10288.5i	0.957326i	−0.877999	+	0.478663i	$0.841122\pi$
$488$	0	0
$489$	−4321.68	−0.399659
$490$	0	0
$491$	−3423.20	−0.314637	−0.157319	−	0.987548i	$-0.550285\pi$
$491$	−3423.20	−0.314637	−0.157319	+	0.987548i	$0.550285\pi$
$492$	0	0
$493$	− 10874.5i	− 0.993436i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10414.2i	− 0.939921i
$498$	0	0
$499$	8210.85	0.736609	0.368305	−	0.929705i	$-0.379938\pi$
$499$	8210.85	0.736609	0.368305	+	0.929705i	$0.379938\pi$
$500$	0	0
$501$	−13382.0	−1.19334
$502$	0	0
$503$	7377.71i	0.653988i	0.945026	+	0.326994i	$0.106036\pi$
$503$	7377.71i	0.653988i	−0.945026	+	0.326994i	$0.893964\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2284.15i	− 0.200084i
$508$	0	0
$509$	456.652	0.0397657	0.0198829	−	0.999802i	$-0.493671\pi$
$509$	456.652	0.0397657	0.0198829	+	0.999802i	$0.493671\pi$
$510$	0	0
$511$	781.781	0.0676790
$512$	0	0
$513$	345.134i	0.0297038i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 697.166i	− 0.0593063i
$518$	0	0
$519$	−4564.43	−0.386043
$520$	0	0
$521$	3000.32	0.252297	0.126148	−	0.992011i	$-0.459738\pi$
$521$	3000.32	0.252297	0.126148	+	0.992011i	$0.459738\pi$
$522$	0	0
$523$	− 7731.71i	− 0.646433i	−0.946325	−	0.323216i	$-0.895236\pi$
$523$	− 7731.71i	− 0.646433i	0.946325	−	0.323216i	$-0.104764\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6713.09i	0.554889i
$528$	0	0
$529$	11848.1	0.973793
$530$	0	0
$531$	204.437	0.0167077
$532$	0	0
$533$	8225.33i	0.668440i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 17116.5i	− 1.37548i
$538$	0	0
$539$	−3083.71	−0.246428
$540$	0	0
$541$	13614.4	1.08194	0.540971	−	0.841041i	$-0.318057\pi$
$541$	13614.4	1.08194	0.540971	+	0.841041i	$0.318057\pi$
$542$	0	0
$543$	− 18309.1i	− 1.44699i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3198.43i	0.250009i	0.992156	+	0.125005i	$0.0398945\pi$
$547$	3198.43i	0.250009i	−0.992156	+	0.125005i	$0.960105\pi$
$548$	0	0
$549$	−286.253	−0.0222531
$550$	0	0
$551$	−357.478	−0.0276390
$552$	0	0
$553$	11443.3i	0.879964i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 178.506i	− 0.0135790i	−0.999977	−	0.00678952i	$-0.997839\pi$
$557$	− 178.506i	− 0.0135790i	0.999977	−	0.00678952i	$-0.00216119\pi$
$558$	0	0
$559$	15368.5	1.16282
$560$	0	0
$561$	4739.24	0.356668
$562$	0	0
$563$	− 10610.5i	− 0.794280i	−0.917758	−	0.397140i	$-0.870003\pi$
$563$	− 10610.5i	− 0.794280i	0.917758	−	0.397140i	$-0.129997\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 6829.34i	− 0.505830i
$568$	0	0
$569$	21286.8	1.56834	0.784171	−	0.620544i	$-0.213088\pi$
$569$	21286.8	1.56834	0.784171	+	0.620544i	$0.213088\pi$
$570$	0	0
$571$	−12064.5	−0.884209	−0.442104	−	0.896964i	$-0.645768\pi$
$571$	−12064.5	−0.884209	−0.442104	+	0.896964i	$0.645768\pi$
$572$	0	0
$573$	− 15674.5i	− 1.14278i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8292.57i	− 0.598309i	−0.954205	−	0.299155i	$-0.903295\pi$
$577$	− 8292.57i	− 0.598309i	0.954205	−	0.299155i	$-0.0967046\pi$
$578$	0	0
$579$	−8761.74	−0.628887
$580$	0	0
$581$	8480.75	0.605578
$582$	0	0
$583$	4.46092i	0 0.000316900i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15928.8i	− 1.12002i	−0.828485	−	0.560011i	$-0.810797\pi$
$587$	− 15928.8i	− 1.12002i	0.828485	−	0.560011i	$-0.189203\pi$
$588$	0	0
$589$	220.679	0.0154379
$590$	0	0
$591$	−22896.3	−1.59362
$592$	0	0
$593$	− 13330.3i	− 0.923116i	−0.887110	−	0.461558i	$-0.847291\pi$
$593$	− 13330.3i	− 0.923116i	0.887110	−	0.461558i	$-0.152709\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4312.61i	0.295650i
$598$	0	0
$599$	3311.48	0.225882	0.112941	−	0.993602i	$-0.463973\pi$
$599$	3311.48	0.225882	0.112941	+	0.993602i	$0.463973\pi$
$600$	0	0
$601$	−6082.46	−0.412826	−0.206413	−	0.978465i	$-0.566179\pi$
$601$	−6082.46	−0.412826	−0.206413	+	0.978465i	$0.566179\pi$
$602$	0	0
$603$	454.664i	0.0307054i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 6162.45i	− 0.412069i	−0.978545	−	0.206035i	$-0.933944\pi$
$607$	− 6162.45i	− 0.412069i	0.978545	−	0.206035i	$-0.0660560\pi$
$608$	0	0
$609$	6909.48	0.459747
$610$	0	0
$611$	−2459.89	−0.162875
$612$	0	0
$613$	8628.65i	0.568529i	0.958746	+	0.284264i	$0.0917493\pi$
$613$	8628.65i	0.568529i	−0.958746	+	0.284264i	$0.908251\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 16309.0i	− 1.06414i	−0.846700	−	0.532070i	$-0.821414\pi$
$617$	− 16309.0i	− 1.06414i	0.846700	−	0.532070i	$-0.178586\pi$
$618$	0	0
$619$	25185.0	1.63533	0.817665	−	0.575694i	$-0.195268\pi$
$619$	25185.0	1.63533	0.817665	+	0.575694i	$0.195268\pi$
$620$	0	0
$621$	2474.60	0.159907
$622$	0	0
$623$	− 8913.75i	− 0.573229i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 155.793i	− 0.00992309i
$628$	0	0
$629$	−26596.4	−1.68596
$630$	0	0
$631$	−13579.0	−0.856689	−0.428344	−	0.903616i	$-0.640903\pi$
$631$	−13579.0	−0.856689	−0.428344	+	0.903616i	$0.640903\pi$
$632$	0	0
$633$	24594.4i	1.54430i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10880.6i	0.676775i
$638$	0	0
$639$	729.053	0.0451344
$640$	0	0
$641$	26562.1	1.63672	0.818362	−	0.574703i	$-0.194883\pi$
$641$	26562.1	1.63672	0.818362	+	0.574703i	$0.194883\pi$
$642$	0	0
$643$	10349.9i	0.634773i	0.948296	+	0.317387i	$0.102805\pi$
$643$	10349.9i	0.634773i	−0.948296	+	0.317387i	$0.897195\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 30738.0i	− 1.86775i	−0.357598	−	0.933875i	$-0.616404\pi$
$647$	− 30738.0i	− 1.86775i	0.357598	−	0.933875i	$-0.383596\pi$
$648$	0	0
$649$	3795.02	0.229534
$650$	0	0
$651$	−4265.38	−0.256795
$652$	0	0
$653$	− 6912.35i	− 0.414244i	−0.978315	−	0.207122i	$-0.933590\pi$
$653$	− 6912.35i	− 0.414244i	0.978315	−	0.207122i	$-0.0664096\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	54.7291i	0.00324990i
$658$	0	0
$659$	3793.27	0.224226	0.112113	−	0.993695i	$-0.464238\pi$
$659$	3793.27	0.224226	0.112113	+	0.993695i	$0.464238\pi$
$660$	0	0
$661$	−30232.2	−1.77897	−0.889483	−	0.456968i	$-0.848935\pi$
$661$	−30232.2	−1.77897	−0.889483	+	0.456968i	$0.848935\pi$
$662$	0	0
$663$	− 16722.0i	− 0.979532i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2563.11i	0.148791i
$668$	0	0
$669$	11767.0	0.680025
$670$	0	0
$671$	−5313.80	−0.305718
$672$	0	0
$673$	8389.60i	0.480528i	0.970708	+	0.240264i	$0.0772340\pi$
$673$	8389.60i	0.480528i	−0.970708	+	0.240264i	$0.922766\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8879.70i	0.504099i	0.967714	+	0.252049i	$0.0811045\pi$
$677$	8879.70i	0.504099i	−0.967714	+	0.252049i	$0.918895\pi$
$678$	0	0
$679$	−10607.6	−0.599530
$680$	0	0
$681$	5451.42	0.306753
$682$	0	0
$683$	− 12555.2i	− 0.703384i	−0.936116	−	0.351692i	$-0.885606\pi$
$683$	− 12555.2i	− 0.703384i	0.936116	−	0.351692i	$-0.114394\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 20443.3i	− 1.13532i
$688$	0	0
$689$	15.7400	0.000870313 0
$690$	0	0
$691$	−30372.7	−1.67211	−0.836057	−	0.548642i	$-0.815145\pi$
$691$	−30372.7	−1.67211	−0.836057	+	0.548642i	$0.815145\pi$
$692$	0	0
$693$	69.8272i	0.00382759i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 14843.2i	− 0.806635i
$698$	0	0
$699$	−683.694	−0.0369953
$700$	0	0
$701$	−33508.8	−1.80543	−0.902716	−	0.430237i	$-0.858430\pi$
$701$	−33508.8	−1.80543	−0.902716	+	0.430237i	$0.858430\pi$
$702$	0	0
$703$	874.303i	0.0469061i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4274.12i	− 0.227362i
$708$	0	0
$709$	−29743.4	−1.57551	−0.787756	−	0.615987i	$-0.788757\pi$
$709$	−29743.4	−1.57551	−0.787756	+	0.615987i	$0.788757\pi$
$710$	0	0
$711$	−801.098	−0.0422553
$712$	0	0
$713$	− 1582.26i	− 0.0831083i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30093.9i	1.56747i
$718$	0	0
$719$	404.733	0.0209930	0.0104965	−	0.999945i	$-0.496659\pi$
$719$	404.733	0.0209930	0.0104965	+	0.999945i	$0.496659\pi$
$720$	0	0
$721$	8818.40	0.455499
$722$	0	0
$723$	− 9561.99i	− 0.491859i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19096.9i	0.974230i	0.873338	+	0.487115i	$0.161951\pi$
$727$	19096.9i	0.974230i	−0.873338	+	0.487115i	$0.838049\pi$
$728$	0	0
$729$	−19193.8	−0.975147
$730$	0	0
$731$	−27733.4	−1.40323
$732$	0	0
$733$	− 28856.0i	− 1.45405i	−0.686610	−	0.727026i	$-0.740902\pi$
$733$	− 28856.0i	− 1.45405i	0.686610	−	0.727026i	$-0.259098\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8440.07i	0.421837i
$738$	0	0
$739$	36005.6	1.79227	0.896134	−	0.443784i	$-0.146364\pi$
$739$	36005.6	1.79227	0.896134	+	0.443784i	$0.146364\pi$
$740$	0	0
$741$	−549.703	−0.0272522
$742$	0	0
$743$	− 36423.7i	− 1.79846i	−0.437475	−	0.899230i	$-0.644127\pi$
$743$	− 36423.7i	− 1.79846i	0.437475	−	0.899230i	$-0.355873\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	593.701i	0.0290795i
$748$	0	0
$749$	9768.79	0.476560
$750$	0	0
$751$	−25845.4	−1.25581	−0.627905	−	0.778290i	$-0.716088\pi$
$751$	−25845.4	−1.25581	−0.627905	+	0.778290i	$0.716088\pi$
$752$	0	0
$753$	− 19125.5i	− 0.925594i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	17862.6i	0.857634i	0.903391	+	0.428817i	$0.141069\pi$
$757$	17862.6i	0.857634i	−0.903391	+	0.428817i	$0.858931\pi$
$758$	0	0
$759$	−1117.03	−0.0534198
$760$	0	0
$761$	−22733.0	−1.08288	−0.541439	−	0.840740i	$-0.682120\pi$
$761$	−22733.0	−1.08288	−0.541439	+	0.840740i	$0.682120\pi$
$762$	0	0
$763$	1991.60i	0.0944965i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 13390.4i	− 0.630379i
$768$	0	0
$769$	23153.6	1.08575	0.542873	−	0.839815i	$-0.317336\pi$
$769$	23153.6	1.08575	0.542873	+	0.839815i	$0.317336\pi$
$770$	0	0
$771$	15613.3	0.729309
$772$	0	0
$773$	26456.0i	1.23099i	0.788141	+	0.615494i	$0.211044\pi$
$773$	26456.0i	1.23099i	−0.788141	+	0.615494i	$0.788956\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16898.9i	− 0.780237i
$778$	0	0
$779$	−487.939	−0.0224419
$780$	0	0
$781$	13533.6	0.620066
$782$	0	0
$783$	− 19891.8i	− 0.907886i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13415.1i	0.607619i	0.952733	+	0.303809i	$0.0982586\pi$
$787$	13415.1i	0.607619i	−0.952733	+	0.303809i	$0.901741\pi$
$788$	0	0
$789$	−15960.2	−0.720151
$790$	0	0
$791$	10627.3	0.477705
$792$	0	0
$793$	18749.3i	0.839605i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31375.6i	1.39445i	0.716850	+	0.697227i	$0.245583\pi$
$797$	31375.6i	1.39445i	−0.716850	+	0.697227i	$0.754417\pi$
$798$	0	0
$799$	4439.04	0.196548
$800$	0	0
$801$	624.013	0.0275261
$802$	0	0
$803$	1015.95i	0.0446479i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 36216.3i	− 1.57977i
$808$	0	0
$809$	−28335.1	−1.23141	−0.615705	−	0.787977i	$-0.711129\pi$
$809$	−28335.1	−1.23141	−0.615705	+	0.787977i	$0.711129\pi$
$810$	0	0
$811$	−34538.1	−1.49543	−0.747717	−	0.664018i	$-0.768850\pi$
$811$	−34538.1	−1.49543	−0.747717	+	0.664018i	$0.768850\pi$
$812$	0	0
$813$	− 32457.1i	− 1.40015i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	911.681i	0.0390400i
$818$	0	0
$819$	246.380	0.0105118
$820$	0	0
$821$	−29834.1	−1.26823	−0.634115	−	0.773239i	$-0.718635\pi$
$821$	−29834.1	−1.26823	−0.634115	+	0.773239i	$0.718635\pi$
$822$	0	0
$823$	36246.8i	1.53522i	0.640919	+	0.767608i	$0.278553\pi$
$823$	36246.8i	1.53522i	−0.640919	+	0.767608i	$0.721447\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 21831.3i	− 0.917955i	−0.888448	−	0.458978i	$-0.848216\pi$
$827$	− 21831.3i	− 0.917955i	0.888448	−	0.458978i	$-0.151784\pi$
$828$	0	0
$829$	−12273.0	−0.514186	−0.257093	−	0.966387i	$-0.582765\pi$
$829$	−12273.0	−0.514186	−0.257093	+	0.966387i	$0.582765\pi$
$830$	0	0
$831$	−17209.3	−0.718391
$832$	0	0
$833$	− 19634.8i	− 0.816693i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	12279.6i	0.507105i
$838$	0	0
$839$	−37441.0	−1.54065	−0.770326	−	0.637650i	$-0.779906\pi$
$839$	−37441.0	−1.54065	−0.770326	+	0.637650i	$0.779906\pi$
$840$	0	0
$841$	−3785.77	−0.155224
$842$	0	0
$843$	− 2410.02i	− 0.0984644i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10890.3i	− 0.441788i
$848$	0	0
$849$	42399.7	1.71396
$850$	0	0
$851$	6268.72	0.252514
$852$	0	0
$853$	1247.81i	0.0500871i	0.999686	+	0.0250436i	$0.00797245\pi$
$853$	1247.81i	0.0500871i	−0.999686	+	0.0250436i	$0.992028\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2927.40i	0.116684i	0.998297	+	0.0583419i	$0.0185813\pi$
$857$	2927.40i	0.116684i	−0.998297	+	0.0583419i	$0.981419\pi$
$858$	0	0
$859$	30892.1	1.22704	0.613518	−	0.789681i	$-0.289754\pi$
$859$	30892.1	1.22704	0.613518	+	0.789681i	$0.289754\pi$
$860$	0	0
$861$	9431.08	0.373299
$862$	0	0
$863$	− 5815.96i	− 0.229406i	−0.993400	−	0.114703i	$-0.963408\pi$
$863$	− 5815.96i	− 0.229406i	0.993400	−	0.114703i	$-0.0365916\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4346.06i	0.170242i
$868$	0	0
$869$	−14871.0	−0.580513
$870$	0	0
$871$	29780.1	1.15851
$872$	0	0
$873$	− 742.589i	− 0.0287890i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4732.26i	0.182209i	0.995841	+	0.0911044i	$0.0290397\pi$
$877$	4732.26i	0.182209i	−0.995841	+	0.0911044i	$0.970960\pi$
$878$	0	0
$879$	4273.60	0.163988
$880$	0	0
$881$	43833.1	1.67625	0.838124	−	0.545480i	$-0.183653\pi$
$881$	43833.1	1.67625	0.838124	+	0.545480i	$0.183653\pi$
$882$	0	0
$883$	37311.7i	1.42201i	0.703185	+	0.711007i	$0.251761\pi$
$883$	37311.7i	1.42201i	−0.703185	+	0.711007i	$0.748239\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44940.5i	1.70119i	0.525822	+	0.850595i	$0.323758\pi$
$887$	44940.5i	1.70119i	−0.525822	+	0.850595i	$0.676242\pi$
$888$	0	0
$889$	−313.867	−0.0118411
$890$	0	0
$891$	8874.98	0.333696
$892$	0	0
$893$	− 145.925i	− 0.00546829i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3941.35i	0.146709i
$898$	0	0
$899$	−12718.8	−0.471854
$900$	0	0
$901$	−28.4038	−0.00105024
$902$	0	0
$903$	− 17621.3i	− 0.649392i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 16241.6i	− 0.594591i	−0.954786	−	0.297295i	$-0.903915\pi$
$907$	− 16241.6i	− 0.594591i	0.954786	−	0.297295i	$-0.0960846\pi$
$908$	0	0
$909$	299.213	0.0109178
$910$	0	0
$911$	2088.21	0.0759444	0.0379722	−	0.999279i	$-0.487910\pi$
$911$	2088.21	0.0759444	0.0379722	+	0.999279i	$0.487910\pi$
$912$	0	0
$913$	11021.1i	0.399500i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 21644.8i	− 0.779469i
$918$	0	0
$919$	40755.3	1.46289	0.731443	−	0.681903i	$-0.238847\pi$
$919$	40755.3	1.46289	0.731443	+	0.681903i	$0.238847\pi$
$920$	0	0
$921$	2016.08	0.0721303
$922$	0	0
$923$	− 47752.3i	− 1.70291i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	617.338i	0.0218728i
$928$	0	0
$929$	24239.7	0.856058	0.428029	−	0.903765i	$-0.359208\pi$
$929$	24239.7	0.856058	0.428029	+	0.903765i	$0.359208\pi$
$930$	0	0
$931$	−645.455	−0.0227217
$932$	0	0
$933$	− 8754.64i	− 0.307196i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 37308.1i	− 1.30075i	−0.759614	−	0.650374i	$-0.774612\pi$
$937$	− 37308.1i	− 1.30075i	0.759614	−	0.650374i	$-0.225388\pi$
$938$	0	0
$939$	−19815.5	−0.688663
$940$	0	0
$941$	109.276	0.00378566	0.00189283	−	0.999998i	$-0.499397\pi$
$941$	109.276	0.00378566	0.00189283	+	0.999998i	$0.499397\pi$
$942$	0	0
$943$	3498.51i	0.120813i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13256.9i	0.454902i	0.973790	+	0.227451i	$0.0730391\pi$
$947$	13256.9i	0.454902i	−0.973790	+	0.227451i	$0.926961\pi$
$948$	0	0
$949$	3584.71	0.122618
$950$	0	0
$951$	9777.29	0.333386
$952$	0	0
$953$	56535.2i	1.92167i	0.277112	+	0.960837i	$0.410623\pi$
$953$	56535.2i	1.92167i	−0.277112	+	0.960837i	$0.589377\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8979.12i	0.303296i
$958$	0	0
$959$	−27393.4	−0.922396
$960$	0	0
$961$	−21939.4	−0.736443
$962$	0	0
$963$	683.870i	0.0228841i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32548.6i	1.08241i	0.840890	+	0.541206i	$0.182032\pi$
$967$	32548.6i	1.08241i	−0.840890	+	0.541206i	$0.817968\pi$
$968$	0	0
$969$	991.976	0.0328863
$970$	0	0
$971$	1604.29	0.0530216	0.0265108	−	0.999649i	$-0.491560\pi$
$971$	1604.29	0.0530216	0.0265108	+	0.999649i	$0.491560\pi$
$972$	0	0
$973$	1210.83i	0.0398945i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11269.0i	− 0.369014i	−0.982831	−	0.184507i	$-0.940931\pi$
$977$	− 11269.0i	− 0.369014i	0.982831	−	0.184507i	$-0.0590688\pi$
$978$	0	0
$979$	11583.7	0.378159
$980$	0	0
$981$	−139.423	−0.00453766
$982$	0	0
$983$	48373.3i	1.56955i	0.619781	+	0.784775i	$0.287221\pi$
$983$	48373.3i	1.56955i	−0.619781	+	0.784775i	$0.712779\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2820.49i	0.0909596i
$988$	0	0
$989$	6536.72	0.210167
$990$	0	0
$991$	792.959	0.0254179	0.0127090	−	0.999919i	$-0.495954\pi$
$991$	792.959	0.0254179	0.0127090	+	0.999919i	$0.495954\pi$
$992$	0	0
$993$	− 38038.1i	− 1.21561i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 52187.1i	− 1.65775i	−0.559430	−	0.828877i	$-0.688980\pi$
$997$	− 52187.1i	− 1.65775i	0.559430	−	0.828877i	$-0.311020\pi$
$998$	0	0
$999$	−48650.4	−1.54077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.m.449.2		6
4.3	odd	2		800.4.c.n.449.5		6
5.2	odd	4		800.4.a.w.1.1	yes	3
5.3	odd	4		800.4.a.u.1.3	✓	3
5.4	even	2	inner	800.4.c.m.449.5		6
20.3	even	4		800.4.a.x.1.1	yes	3
20.7	even	4		800.4.a.v.1.3	yes	3
20.19	odd	2		800.4.c.n.449.2		6
40.3	even	4		1600.4.a.cq.1.3		3
40.13	odd	4		1600.4.a.ct.1.1		3
40.27	even	4		1600.4.a.cs.1.1		3
40.37	odd	4		1600.4.a.cr.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.u.1.3	✓	3	5.3	odd	4
800.4.a.v.1.3	yes	3	20.7	even	4
800.4.a.w.1.1	yes	3	5.2	odd	4
800.4.a.x.1.1	yes	3	20.3	even	4
800.4.c.m.449.2		6	1.1	even	1	trivial
800.4.c.m.449.5		6	5.4	even	2	inner
800.4.c.n.449.2		6	20.19	odd	2
800.4.c.n.449.5		6	4.3	odd	2
1600.4.a.cq.1.3		3	40.3	even	4
1600.4.a.cr.1.3		3	40.37	odd	4
1600.4.a.cs.1.1		3	40.27	even	4
1600.4.a.ct.1.1		3	40.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.25747i q^{3} +9.15590i q^{7} -0.640965 q^{9} +O(q^{10})\)	\(q-5.25747i q^{3} +9.15590i q^{7} -0.640965 q^{9} -11.8984 q^{11} +41.9826i q^{13} -75.7604i q^{17} -2.49047 q^{19} +48.1368 q^{21} +17.8566i q^{23} -138.582i q^{27} +143.538 q^{29} -88.6094 q^{31} +62.5556i q^{33} -351.059i q^{37} +220.722 q^{39} +195.922 q^{41} -366.067i q^{43} +58.5931i q^{47} +259.169 q^{49} -398.308 q^{51} -0.374917i q^{53} +13.0936i q^{57} -318.952 q^{59} +446.597 q^{61} -5.86861i q^{63} -709.343i q^{67} +93.8805 q^{69} -1137.43 q^{71} -85.3855i q^{73} -108.941i q^{77} +1249.83 q^{79} -745.895 q^{81} -926.261i q^{83} -754.648i q^{87} -973.552 q^{89} -384.389 q^{91} +465.861i q^{93} +1158.55i q^{97} +7.62648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 36 q^{9}+O(q^{10}) \) 6 * q - 36 * q^9	\( 6 q - 36 q^{9} - 62 q^{11} + 174 q^{19} - 140 q^{21} - 168 q^{29} - 588 q^{31} - 64 q^{39} - 690 q^{41} - 718 q^{49} - 2702 q^{51} + 2080 q^{59} + 964 q^{61} + 1228 q^{69} - 4096 q^{71} + 1996 q^{79} - 1098 q^{81} - 2378 q^{89} - 8064 q^{91} + 3908 q^{99}+O(q^{100}) \) 6 * q - 36 * q^9 - 62 * q^11 + 174 * q^19 - 140 * q^21 - 168 * q^29 - 588 * q^31 - 64 * q^39 - 690 * q^41 - 718 * q^49 - 2702 * q^51 + 2080 * q^59 + 964 * q^61 + 1228 * q^69 - 4096 * q^71 + 1996 * q^79 - 1098 * q^81 - 2378 * q^89 - 8064 * q^91 + 3908 * q^99

Introduction

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